Please join us for:

a reception on Friday, November 3rd, from 8:00 - 10:30 p.m.
in the lounge on the 13th floor of the Courant Institute.

a dinner on Saturday, November 4th, from 7:00 - 11:00 p.m.
in the John Ben Snow dining room on the top floor of Bobst Library.
The cost is $50 per person and it includes dinner and drinks. Reservations
and payment must be made at the check-in table before noon on Friday.

All conference participants are invited to attend. We hope you'll be
able to join us!

Conference Schedule

Titles & Abstracts

Richard Bass
Title: Symmetric jump processes
Abstract: We consider reversible Markov chains on Z^d where the jump probabilities are comparable to those of a symmetric stable process. We will discuss tightness estimates, Harnack inequalities, heat kernel bounds, and central limit theorems.

Gerard Ben Arous
Title: Aging in a random landscape
Abstract: Aging is a transient mechanism, present in many stochastic dynamics in random systems. We will present a discussion of the possible various scenario of aging proposed by physicists (Bouchaud, Mézard), and a survey of mathematical results for the simplest models: Spherical Sherrington-Kikpatrick model (joint work with A.Dembo and A.Guionnet), Derrida's Random Energy Model (joint work with A.Bovier and V.Gayrard).

Maury Bramson
Title: Nontranslation Invariant Equilibria for One-Dimensional Exclusion Processes
Abstract: The equilibrium measures for the nearest neighbor exclusion process in one dimension are well understood. When the transition kernel is translation invariant with a nonzero drift, there exist nontranslation invariant equilibria in addition to the translation invariant product measures. Here, we report on recent work with Tom Mountford, where it is shown that the same is true when the nearest neighbor assumption is replaced by the assumption that the range of the transition kernel is finite.

Eric Carlen
Title: Spectral Gap for the Kac Walk with Maxwellian Collisions
Abstract: This talk presents joint work with M. Carvalho and M. Loss. We prove bounds on the rate of convergence to equilibrium for an N particle system of three-dimensional particles undergoing momentum and energy conserving collisions. This is a more physical analog of a model originally introduced by Kac, concerning one-dimensional particles with energy conservation only. The later problem was solved only recently by Janvresse (published) and Maslin (unpublished). The solution by Janvresse uses Yau's martigale method, and thus uses spectral estimates on certain conditional expectation operators in the Dirichlet form associated to the dynamics. We rely on the analysis of the same conditional expectation operators, but show that it is not necessary to analyze their spectrum in the Dirichlet form associated to the dynamics. Indeed, we show that a simple purely geometric correlation estimate, independent of the dynamics, relates the rate for N particles to the rate for N-1, and thus one need only work out the spectrum of the Dirichlet form for N=2. This leads sharp bounds for the original Kac model, and things are simple enough that the method extends to the case of physical collisions described above.

Bernard Derrida
Title: Large deviations in 1 D exclusion process
Abstract: Two main exact results will be presented:
For the fully asymmetric exclusion process with P particles on a ring of N sites, one can derive an exact expression for the large deviation function [1] of the time averaged current. For large N, this large deviation function takes a universal shape independent of the density and which should characterize a large class of growing interface models [2].

For the symmetric exclusion process on an open interval, where particles are injected at the left and are ejected at the right, one can find an expression in the steady state for the probability distribution of an arbitrary density profile.

[1] B. Derrida, J.L. Lebowitz, "Exact large deviation function in the asymmetric exclusion process" Phys. Rev. Lett. 80, 209-213 (1998)
[2] B. Derrida, C. Appert, "Universal large deviation function of the Kardar-Parisi-Zhang equation in one dimension" J. Stat. Phys. 94, 1-30 (1999)

David Jerison
Title: The hot spots conjecture of J. Rauch
Abstract: In 1974 Jeff Rauch conjectured that the point of highest temperature in an insulated region (that is, one with zero Neumann data) ``usually'' tends to the boundary as time tends to infinity. This is equivalent to the assertion that the first nonconstant Neumann eigenfunction achieves its maximum on the boundary. There has been recent progress on this problem using probabilistic methods by Ba\~nuelos and Burdzy, Burdzy and Werner, and Bass and Burdzy. We will survey these results and closely related results developed by the speaker in joint work with N. Nadirashvili using methods from elliptic PDE.

Harry Kesten
Title: A large deviation result for the range of random walk and for the Wiener sausage
Abstract: Let S(n), n = 0, 1, 2, ... be a random walk on the d-dimensional integer lattice. Let R(n) be its range, that is, the number of distinct points among 0, S(1), ...,S(n-1). We prove that C(x):= lim (-1/n) log P{R(n) >= nx} exists and derive some convexity and monotonicity properties of C(x).

We also prove an anologous result for the Wiener sausage. Let B(t) be a standard d-dimensional; Brownian motion, and let V(t) be the volume swept out by B(s) + A, 0 <= s <= t, for some fixed d-dimensional set A. Then D(x) := lim (-1/t) log P{V(t) >= tx} exists and has convexity and momotonicity properties similar to those of C(x).

This is joint work with Yuji Hamana.

Shigeo Kusuoka
Title: Malliavin Calculus revisited
Abstract: In 1987, Stroock and I prove the following regularity estimates for diffusion semigroups under the Uniform H\"ormander condition by using Malliavin Calculus.

$$
\parallel V_{[\alpha_1]}\cdots V_{[\alpha_k]}
P_t^c V_{[\alpha_{k+1}]}\cdots V_{[\alpha_{k+\ell}]}f
\parallel_{L^p(dx)}
$$
$$
\leq C t^{-(\parallel \alpha_1 \parallel + \cdots
\parallel \alpha_{k+\ell}\parallel )/2}
\parallel f \parallel_{L^p(dx)},
\quad f \in C_0^{\infty}({\bf R}^N)
$$
for any $p\in [1,\infty ].$

This result is strongly related to a subellpticity condition by Rothschild and Stein. We show that these estimates hold even if there is no hypoellipticity. Actually we can show these estimate under the condition that a certain Lie algebra generated by vector fields is finitely generated as a $C^{\infty}_b$-module.

Paul Malliavin
Title: Probability measure on the group of diffeomorphism of the circle
Abstract: Realization of the heat process for the canonic kahler structure on the homogeneous space Diff(S)/S

Stefano Olla
Title: Tagged particles and self-diffusion.
Abstract: I will review some central limit theorems for tagged particles in exclusion processes and related problems: from the symmetric (reversible) case studied by Kipnis and Varadhan till recent results on the asymmetric case by Sethuraman, Varadhan and Yau. Then I will show the connection with the fluctuation- dissipation theorem (bulk-diffusion) by Landim and Yau. Finally I will present some new results concerning the smooth dependence on the density of particles of the self (and bulk)-diffusion effective matrix, and on its stability with respect to approximations by finite exclusion processes in large periodic lattices (joint work with Landim and Varadhan).

George Papanicolaou
Title: Variational principles for convection, diffusion and flows and applications.
Abstract: I will describe how saddle-point variational principles can be used to study the effective diffusivity for convection-diffusion at large Peclet number. I will also show how saddle-point variational principles can be used to bound the behavior of the eddy viscosity of cellular flows at high Reynolds number.

Jeremy Quastel
Title: Some degenerate diffusions arising in interacting particle systems
Abstract: We will review some results on well-posedness of the martingale problem for diffusions whose coefficients are neither very regular, nor uniformly elliptic. We will also describe how the problem arose in the study of tagged particles in an interacting system.

Fraydoun Rezakhanlou
Title: A stochastic model for a moderately dense gas and its kinetic limit
Abstract: Boltzmann equation provides a successful description for dilute gases and can be derived from a Hamiltonian system in a suitable scaling limit. This description is no longer valid when the density of the gas increases. In 1922, Enskog proposes a modification of the Boltzmann equation to explain the dynamical behavior of the density profile of a moderately dense gas. It is not known whether the Enskog equation can be derived from a Hamiltonian system. Recently Mario Pulvirenti and I have studied a stochastic particle system in ${\Bbb R}^d$ as a microscopic model for a moderately dense gas. In this model there are $N$ particles that travel freely between stochastic collisions. We regard each particle as a ball of radis $\sigma$ and with probability $\epsilon$ an elastic collision occurs when two particles collide. With the remaining probability $1-\e$ the particles pass each other. We show that if $N$ goes to infinity and ${\epsilon}^dN$ goes to a finite positive number, then the microscopic particle density converges to a solution of an Enskog equation.

Thomas Spencer
Title: Universality and the Ising model in two dimensions
Abstract: We shall describe joint work with Haru Pinson on phase transitions for 2 dimensional Ising models. We prove that for small perturbations of the nearest neighbor Ising model, critical exponents associated with specific heat and the correlation length are independent of the perturbation. We also discuss critical phenomena for a coupled chain of anharmonic oscillators.

Herbert Spohn
Title: Current fluctuations in the asymmetric simple exclusion process
Abstract: We consider the totally asymmetric simple exclusion process in one dimension with step Bernoulli initial conditions. The quantity of interest is the particle current through the origin, which we map onto a last passage percolation problem with boundary conditions encoding the initial measure. Based on results of Baik and Rains this allows us to show that the typical fluctuations of the time-integrated current are of the size t^{1/3} and to discuss how its distribution on that scale depends on the initial conditions.

Alain Sznitman
Title: Some recent results on random walks in random environment
Abstract: Random walks in random environment are one of the basic models of random motions in a random medium. However their asymptotic behavior is so far rather poorly understood when the dimension is bigger than one. We describe in this talk some of the recent progresses concerning certain multi-dimensional walks with "ballistic behavior".

V. S. Varadarajan
Title: Some mathematical reminiscences
Abstract: I shall talk about some of the instances when I collaborated with Professor Varadhan. Incidentally this will allow me to sketch the nature of the life and mathematics in Calcutta in the early 1960's, at least in an impressionistic manner.

Ruth Williams
Title: Dynamic control of stochastic networks and reflected diffusions
Abstract: This talk will illustrate how reflected diffusions can arise as asymptotically optimal solutions to dynamic control problems for stochastic networks.

Hotels for the Conference

Allerton Hotel
302 West 22nd Street
212-243-6017

Carlton Arms Hotel
160 East 25th Street
212-679-0680

Chelsea Hotel
222 West 23rd Street
212-243-3700

Club Quarters
52 William Street
212-269-6400

Comfort Inn
42 West 35th Street
212-947-0200

Gramercy Park Hotel
Two Lexington Avenue & 21st Street
212-475-4320

Herald Square
19 West 31st Street
212-279-4017

Holiday Inn Downtown
138 Lafayette Street
212-966-8898

Hotel Metro
45 West 35th Street
212-947-2500

Hotel Seventeen
225 East 17th Street
212-475-2845

Hotel Wolcott
Four West 31st Street
212-268-2900

Madison Hotel
21 East 27th Street & Madison Avenue
212-253-7373

Manhattan Hotel
17 West 32nd Street
212-736-1600

Murray Hill Inn
143 East 30th Street
212-545-0879

Pickwick Arms Hotel
230 East 51st Street
212-355-0300

Stanford Hotel
43 West 32nd Street
212-563-1500

Washington Square Hotel
103 Waverly Place
212-777-9515